Basically, what happens is with a closed fist, one lights a lighter into the fist and when the hand is opened, a flame appears. It's actually a lot easier than it looks, for those who aren't afraid of fire.

Disclaimer: If you end up burning your hand to ashes and cooked meat, it's not my fault.

1. Make a closed fist with one hand (for this example, we will have the lighter in the right and the closed hand on the left) leaving a space inside the palm area. To make sure you've got this part right, stick out your right index finger and grab it with the left hand. Make sure your right index finger doesn't stick out the end of your closed hand. Remove your right hand from your left.
2. Put the lighter near the closed hand, pointing the end where the fire comes out into the space in the left hand.
3. With your right thumb, press down on the button that allows gas to leave the lighter. Direct this gas into the space in your closed left hand.
4. After about 5-10 seconds of releasing the gas, light the lighter. Allow the gas to set fire inside your left hand for half a second before opening your left hand to see a flame emerge.
5. Impress people who are easily impressed.

Okay, shrimpo's write-up is really exciting - especially if you enjoy the occasional sarcastic "wow" and people laughing at you, not your trick.

If you really want to shock people, I have a much more effecting method of making a flame appear in your hand. Now, be warned, this is very dangerous, but where are you ever going to get in life - or more importantly, at parties - unless you take a few risks?

For starters, there are a few things you'll need:

As with shrimpo's method, you'll need a lighter. I recommend a jet or a similar lighter. Not because it will light better, but because it looks really cool.

You'll also need some flamable liquid, such as petroleum (or gasoline to Americans). Only about a cup is required.

While dipping your hand in gasoline is one way to make flame appear from it another more mundane method is to use alcohol. What you'll need is a can of either strong beer or liquor and a lighter or match.

First, while nobody is looking splash your (alcoholic) drink over you hand. Then light it with you lighter and wave it around like a maniac (but not fast enough to put the fire out). From here you can either pretend your in horrible pain or play it cool, but it being alcohol it doesn't generate very much heat.

Another alternate idea is to pretend that nothing is happening while making sure everybody can see your hand. For best results do this in a darkened room or at night as alcohol flames aren't that bright.

In reply to the numerous comments saying that this won't work with beer, I personally have done this on many occasions. Strong beer is not even necessary if you're lucky, but it works best. The alcohol does light up and noticably too. However I won't recommend that you try it just to make sure it works though, as it might be riskier than I think it is. Just ask yourself what's more important to you, no risk of bandages or oohs and aahs.

I'm also told that doing this with spirits or pure alcohol could get you a trip to the emergency ward if it runs between you fingers, so be careful and start with small amounts first! BTW, if the above writeup about gasoline is still around please realise it's a joke. Even alcohol burns pretty hot, so just forget about petrol.

While you are fire breathing or other such impressive feats, put a small amount of paraffin on your hand (the back of your hand is easier) and light it. Unlike alcohol which burns with a blueish flame, paraffin burns with a bright yellow flame.

For further impressive ideas, if you tap the burning part of your hand (quickly!) - as if to pat the fire out - then you can carry some of the paraffin over to your other hand and make it look as if the fire is spreading.

It has been pointed out that, if you had a mouthful of paraffin at this point you could spray flame, using your hand as the ignitor.

I have not tried this and might result in harm... only to be tried near a handy hosepipe or swimming pool...

I have actually lit my hand on fire with alcohol twice before (once accidentally, and then again on purpose, because it looked really cool), and feel that I should probably add a few comments here. The first has to do with the comment about flammable beer. As far as I know, alcohol below 100 proof (50%) will not burn. I actually heard that that was why it is called "100 proof", because when they were first figuring out everything about alcohol, they discovered that it would only burn once they had gotten it to a certain point, and thinking that to be pure, decided that it would be 100 proof. Don't quote me on that, though. It's quite possible that it has very little to do with the real story.1

Anyway, onto my story. What happened was that I had a candle and a bottle of rubbing alcohol (91%, if you're curious) nearby, completely coincidental. There was an ice storm going on, and our power was out. I needed this candle lit. There was, however, no wick protruding from the wax, and, being the impatient and impulsive fellow that I am, spied the alcohol on the counter and figured I could simply burn away a little of the wax. I poured a little in the cap, meaning to then dump that in the candle, but the liquid sloshed and poured out all over my hand, completely covering it. To this I never gave a second thought. I went over to the candle, poured the capful of alcohol in, and went and got a match. And struck it. With a hand covered in alcohol. I believe the exact thought that went through my mind as I saw my hand burst into flame was this: "OH, SHIT!" I will also say this: it is amazing how quickly a person can move when they are on fire. I waved my hand so quickly that the fire actually went out, quite a feat when you are dealing with not a tiny alcohol fire.

As I said before, the power was out, and it was quite dark, meaning that it looked really cool seeing my hand burning with that bright blue flame. If you have ever seen the movie Hellboy or one of the trailers for it, you will know what I'm talking about. It looked exactly like that.

So I promptly lit my hand on fire again. This time I did it more carefully: I stood next to the sink. And it was just as pretty the second time. I did learn something new thing this time around, though. If you light your hand on fire and then put it out really quickly, while there is still alcohol to burn, it does not hurt. If, however, you wait even one or two seconds to watch it burn before putting it out, it gets very, very, hot. On fire, in fact. So a word to the wise: if you're planning on doing this, light your hand on fire and then put it out IMMEDIATELY. Your smooth, flesh-colored skin will really appreciate it, and you can escape without harm.2

I was not actually hurt. I did manage to put it out the second time quickly enough to escape with no burns, but only by virtue of the sink that I had strategically positioned next to me. It did hurt like a monkey for a few minutes, though. The one lasting effect of these little adventures was that all the hair on my hand was burned off, and my hands looked very strange sitting next to each other for about two or three months after that.

Oh and I would not recommend this method to get the wick out. It worked, but then the next day, when I picked up the candle, the glass cup candle holder thing it was in shattered at the first touch, spraying glass shards all around my kitchen. Which is I now restrict my glass-shattering to other people's houses.

1Zerotime informs me that he has done the same thing with Finlandia vodka, with about 34% alcohol. Maybe I got that story from a leper... (not that there's anything wrong with that)
In addition I learned from the proof node that the term does come from testing for the alcohol content, but what they actually did was mix the alcohol with gunpowder, and if it exploded, there was "proof" that it was good.

2futilelord tells me that most rubbing alcohol you buy at a store is about 50% alcohol, and with that you can keep your hand burning for about 20 seconds without lasting damage, if you can take the pain. He also says the flames weren't that great, though, so if you're going for effect (and you probably are, if you're lighting your hand on fire), I might recommend the higher alcohol content for a shorter time.

His grand finale would be to secretly plunge his hands into water, and then into benzine. He would then brush a hand past one of the bunsen burners he had going on either side of his table and scream at the sight of the flames. He'd try to clap the burning hand out, only for his other hand to catch alight. At this point he'd chase the hysterical youngsters out into the street and that would be the end of the show. Supposedly the water keeps the benzine floating on top of it moving quickly enough that one doesn't get hurt.

He then goes on to describe trying to convince his university buddies that he had done this unharmed. He repeated the trick for them, realising too late that in the years since he'd last done the trick he'd grown hairs on the backs of his hands, each of which functioned as a tiny little wick to deliver the heat to his skin. He finishes by mentioning that after that, he could perform the trick with impunity because he no longer had any hair on his hands...